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Keywords:
equivalent martingale measure; option pricing; stochastic processes; non-standard analysis
equivalent martingale measure; option pricing; stochastic processes; non-standard analysis
Summary:
The concept of an equivalent martingale measure is of key importance for pricing of financial derivative contracts. The goal of the paper is to apply infinitesimals in the non-standard analysis set-up to provide an elementary construction of the equivalent martingale measure built on hyperfinite binomial trees with infinitesimal time steps.
References:
[1] Albeverio S., Fenstad J.E., Hoegh-Krohn R., Lindstrom T.: Nonstadard Methods in Stochastic Analysis and Mathematical Physics. Dover Publications, 1986. MR 0859372
[2] Anderson R.M.: A nonstandard representation for Brownian motion and Itô integration. Israel Math. J. 25 (1976), 15–46. DOI 10.1007/BF02756559 | MR 0464380 | Zbl 0327.60039
[3] Bacheier L.: La Theorie de la Speculation. Annales de l'Ecole Normale Superieure, 3, Gauthier-Villars, Paris, 1900. MR 1508978
[4] Berg I.: Principles of Infinitesimal Stochastic and Financial Analysis. World Scientific, Singapore, 2000. MR 1789967 | Zbl 0964.91024
[5] Brown R.: A brief account of microscopical observations made in the months of June, July and August, 1827, on the particles contained in the pollen of plants; and on the general existence of active molecules in organic and inorganic bodies. Phil. Mag. 4 (1828), 161–173. DOI 10.1080/14786442808674769
[6] Cox J., Ross S., Rubinstein M.: Option pricing: a simplified approach. J. Financial Econom. 7 (1979), 229–263. DOI 10.1016/0304-405X(79)90015-1 | Zbl 1131.91333
[7] Cutland N., Kopp E., Willinger W.: A nonstandard approach to option pricing. Mathematical Finance 1 (1991), no. 4, 1–38. DOI 10.1111/j.1467-9965.1991.tb00017.x | Zbl 0900.90104
[8] Cutland N., Kopp E., Willinger W.: A nonstandard treatment of options driven by Poisson processes. Stochastics Stochastics Rep. 42 (1993), 115–133. DOI 10.1080/17442509308833813 | MR 1275815 | Zbl 0808.90018
[9] Cutland N., Kopp E., Willinger W.: From discrete to continuous stochastic calculus. Stochastics Stochastics Rep. 52 (1995), 173–192. DOI 10.1080/17442509508833970 | MR 1381667 | Zbl 0864.60039
[10] Cutland N., Kopp E., Willinger W., Wyman M.C.: Convergence of Snell envelopess and critical prices in the American put. Mathematics of Derivative Securitities (M.A.H. Dempster and S.R. Pliska, eds.), Cambridge University Press, Cambridge, 1997, pp. 126–140. MR 1491372
[11] Cutland N.: Loeb Measures in Practice: Recent Advances. Lecture Notes in Mathematics, 1751, Springer, Berlin, 2000. DOI 10.1007/b76881 | MR 1810844 | Zbl 0963.28015
[12] Einstein A.: Investigations on the theory of the Brownian movement. R. Fürth (Ed.), Dover Publications, New York, 1956. MR 0077443 | Zbl 0936.01034
[13] Herzberg F.S.: Stochastic Calculus with Infinitesimals. Springer, Heidelberg, 2013. MR 2986409 | Zbl 1302.60006
[14] Hull J.: Options, Futures, and Other Derivatives. 8th edition, Prentice Hall, 2011. Zbl 1087.91025
[15] Hurd A.E., Loeb P.A.: An Introduction to Nonstandard Real Analysis. Academic Press, Orlando, FL, 1985. MR 0806135 | Zbl 0583.26006
[16] Kalina M.: Probability in the alternative set theory. Comment. Math. Univ. Carolin. 30 (1989), no. 2, 347–356. MR 1014134 | Zbl 0677.03039
[17] Keisler J.H.: An infinitesimal approach to stochastic analysis. Mem. Amer. Math. Soc. 297, 1984. MR 0732752 | Zbl 0529.60062
[18] Keisler J.H.: Elementary Calculus - an Infinitesimal Approach. 2nd edition, University of Wisconsin - Creative Commons., 2000. Zbl 0655.26002
[19] Kopp P.E.: Hyperfinite Mathematical Finance. in Arkeryd et al. Nonstandard Analysis: Theory and Applications, Kluwer, Dordrecht, 1997. MR 1603237 | Zbl 0952.91028
[20] Lindstrom T.: Nonlinear stochastic integrals for hyperfinite Lévy processes. Logic and Analysis 1 (2008), 91–129. DOI 10.1007/s11813-007-0004-7 | MR 2403500 | Zbl 1156.60035
[21] Loeb P.A.: Conversion from nonstandard to standard measure spaces and applications in probability theory. Trans. Amer. Math. Soc. 211 (1975), 113–122. DOI 10.1090/S0002-9947-1975-0390154-8 | MR 0390154 | Zbl 0312.28004
[22] Nelson A.: Internal set theory: a new approach to nonstandard analysis. Bull. Amer. Math. Soc. 83 (1977), no. 6, 1165–1198. DOI 10.1090/S0002-9904-1977-14398-X | MR 0469763 | Zbl 0373.02040
[23] Nelson E.: Radically Elementary Probability Theory. Princeton University Press, Princeton, NJ, 1987. MR 0906454 | Zbl 0651.60001
[24] Pudlák P., Sochor A.: Models of the alternative set theory. J. Symbolic Logic 49 (1984), 570–585. DOI 10.2307/2274190 | MR 0745386 | Zbl 0578.03028
[25] Rebonato R.: Volatility and Correlation: the Perfect Hedger and the Fox. 2nd edition, Wiley, 2004.
[26] Robinson A.: Nonstandard analysis. Proc. Roy. Acad. Amsterdam Ser. A 64 (1961), 432–440. MR 0142464 | Zbl 0424.01031
[27] Shreve S.: Stochastic Calculus for Finance II. Continuous Time Models. Springer, New York, 2004. MR 2057928 | Zbl 1068.91041
[28] Shreve S.: Stochastic Calculus for Finance I. The Binomial Asset Pricing Model. Springer, New York, 2005. MR 2049045 | Zbl 1068.91040
[29] Vopěnka P.: Mathematics in the Alternative Set Theory. Teubner, Leipzig, 1979. MR 0581368
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